Book cover for University Calculus: Early Transcendentals

University Calculus: Early Transcendentals

Joel Hass, Maurice D. Weir, George B. Thomas, Jr.

ISBN #9780321999580

3rd Edition

6,517 Questions

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42,715 Students Helped

Homework Questions

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Summary
Learning Objectives
Key Concepts
Example Problems
Explanations
Common Mistakes

Summary

This section emphasizes the critical role functions play in calculus and real-world modeling. Functions are defined as rules connecting inputs with unique outputs and can be represented in various forms such as graphs, formulas, tables, and words. Understanding domains, ranges, and properties like increasing/decreasing behavior and symmetry is essential. Additionally, combining functions and applying transformations are key techniques that broaden the application of functions in mathematics.

Learning Objectives

1

Understand the definition of a function and be able to identify its domain and range.

2

Analyze various representations of functions including equations, graphs, numerical tables, and verbal descriptions.

3

Examine properties of functions such as increasing/decreasing behavior, symmetry (even/odd), and the implications of the vertical line test.

4

Learn how to combine and transform functions through operations like addition, subtraction, multiplication, and division.

Key Concepts

CONCEPT

DEFINITION

Function

A rule that assigns each element x in a set D (domain) a unique element f(x) in a set Y (range).

Domain

The set of all possible input values (x-values) for which the function is defined.

Range

The set of all output values (f(x)) that result from using the function on its domain.

Graph of a Function

The set of ordered pairs (x, f(x)) plotted in the Cartesian plane representing the behavior of the function.

Vertical Line Test

A test used to determine if a graph represents a function; a vertical line should intersect the graph at no more than one point for each x-value.

Piecewise-Defined Function

A function that is defined by different formulas over different parts of its domain.

Even Function

A function f(x) that satisfies f(-x) = f(x) for every x in its domain, indicating symmetry about the y-axis.

Odd Function

A function f(x) that satisfies f(-x) = -f(x) for every x in its domain, indicating symmetry about the origin.

Function Operations

The processes of adding, subtracting, multiplying, and dividing functions to create new functions.

Example Problems

Example 1

Find the domain and range of each function. $$f(x)=1+x^{2}$$

Example 2

Find the domain and range of each function. $$f(x)=1-\sqrt{x}$$

Example 3

Find the domain and range of each function. $$F(x)=\sqrt{5 x+10}$$

Example 4

Find the domain and range of each function. $$g(x)=\sqrt{x^{2}-3 x}$$

Example 5

Find the domain and range of each function. $$f(t)=\frac{4}{3-t}$$
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Step-by-Step Explanations

QUESTION

Given the function y = x², determine its natural domain and range.

STEP-BY-STEP ANSWER:

Step 1: Recognize that the formula x² is defined for every real number x.
Step 2: Conclude that the natural domain is all real numbers (-∞, ∞).
Step 3: Notice that squaring any real number results in a nonnegative output.
Step 4: Conclude that the range is all nonnegative real numbers, [0, ∞).
Final Answer: Domain is (-∞, ∞) and range is [0, ∞).

Determining Domain and Range for y = x²

QUESTION

Determine whether the circle defined by x² + y² = 1 represents a function of x.

STEP-BY-STEP ANSWER:

Step 1: Visualize or draw the circle x² + y² = 1 in the Cartesian plane.
Step 2: Imagine drawing vertical lines (lines parallel to the y-axis) through various x-values.
Step 3: Observe that some vertical lines intersect the circle at two points.
Step 4: Since the vertical line test fails (there is more than one y-value for some x's), the graph is not a function.
Final Answer: The graph of x² + y² = 1 is not a function of x.

Applying the Vertical Line Test

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Common Mistakes

  • Confusing the domain (input values) with the range (output values) of a function.
  • Failing to properly restrict the domain when a function’s formula does not work for all real numbers (e.g., division by zero or square roots of negative numbers).
  • Misapplying the vertical line test by not checking every possible vertical line.
  • Assuming a graph is a function without verifying that each x-value corresponds to exactly one y-value, especially when dealing with piecewise or nonstandard graphs.